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Topic: Cauchy sequence

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Complete space

Limit of a sequence

Totally bounded

Construction of real numbers

Tensor product

Uniformly continuous

Banach space

Real number

Cauchy product

Open set

Continuous

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	Sequence - Wikipedia, the free encyclopedia
		A finite sequence is also called an n-tuple.
		A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.
		If the terms of the sequence are a subset of a ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing.
	en.wikipedia.org /wiki/Sequence (631 words)

	Cauchy sequence - Wikipedia, the free encyclopedia
		In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses.
		Cauchy sequences require the notion of distance so they can only be defined in a metric space.
		They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence.
	www.wikipedia.org /wiki/Cauchy_sequence (604 words)

	Encyclopedia: Cauchy sequence (Site not responding. Last check: 2007-10-07)
		In 1833 the deposed king Charles X of France summoned Cauchy to be tutor to his grandson, the duke of Bordeaux, an appointment which enabled Cauchy to travel and thereby become acquainted with the favourable impression which his investigations had made.
		Returning to Paris in 1838, Cauchy refused a proffered chair at the CollÃ¨ge de France, but in 1848, the oath having been suspended, he resumed his post at the Ãcole Polytechnique, and when the oath was reinstituted after the coup d'Ã©tat of 1851, Cauchy and FranÃ§ois Arago were exempted from it.
		Cauchy had two brothers: Alexandre Laurent Cauchy (1792â“1857), who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849; and EugÃ¨ne FranÃ§ois Cauchy (1802â“1877), a publicist who also wrote several mathematical works.
	www.nationmaster.com /encyclopedia/Cauchy-sequence (652 words)

	Talk:Cauchy sequence - Wikipedia, the free encyclopedia
		Cauchy net redirects here, yet there seems to be nothing about the concept here....
		Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to converge.
		All Cauchy sequences of real or complex numbers converge, hence testing that a sequence is Cauchy is a test of convergence.
	en.wikipedia.org /wiki/Talk:Cauchy_sequence (393 words)

	Real number
		The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use.
		The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance.
		If we have a space where Cauchy sequences are meaningful (such as a metric space, i.e., a space where distance is defined, or more generally a uniform space), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion).
	www.ebroadcast.com.au /lookup/encyclopedia/re/Real_number.html (2256 words)

	Biogrpahy of Cauchy (Site not responding. Last check: 2007-10-07)
		Cauchy was the first to make a rigorous study of the conditions of convergence of infinite series in addition to his definition of an integral.
		Cauchy proved that the angles of a complex polyhedron are determined by its faces.
		Cauchy is best known for his work with the convergence and divergence of the infinite series and his work with the complex series.
	www.andrews.edu /~calkins/math/biograph/199900/biocauch.htm (1447 words)

	[No title]
		This quiz is designed to test your knowledge of concepts in sequences such as convergence, Comment.
		The limit points of the sequence a_n = (-1)^n n / (2n+1) for n = 0, 1, 2, 3,...
		The inf of the sequence a_n = 1 + (-1)^n / n for n = 1,2,3,...
	www.math.ucla.edu /~tao/java/MultipleChoice/sequences.txt (989 words)

	PlanetMath: if $d(x_i, x_{i+1})<1/2^i$ then $x_i$ is a Cauchy sequence (Site not responding. Last check: 2007-10-07)
		, is a sequence in a metric space.
		is a Cauchy sequence" is owned by matte.
		is a Cauchy sequence, born on 2004-09-22, modified 2004-09-22.
	planetmath.org /encyclopedia/IfDx_iX_i112iThenX_iIsACauchySequence.html (93 words)

	Cauchy sequence (Site not responding. Last check: 2007-10-07)
		In mathematical analysis, a Cauchysequence is a sequence whose terms become arbitrarily close to each otheras the sequence progresses.
		They are of interest because,given certain conditions, all such sequences converge to a limit, and one cantest for "Cauchiness" without having the value of the actual limit.
		Roughly speaking, the terms of the sequence are getting closer and closer together in a way thatsuggests that the sequence ought to have a limit in M.Nonetheless, this may not be the case.
	www.therfcc.org /cauchy-sequence-32807.html (268 words)

	Cauchy sequence: Definition and Links by Encyclopedian.com - All about Cauchy sequence (Site not responding. Last check: 2007-10-07)
		in a metric space (M, d) is called a Cauchy sequence (or Cauchy for short) if for every positive real number r, there is an integer N such that for all integers m and n greater than N the distance d(x
		A metric space in which every Cauchy sequence has a limit is called complete.
		See Complete space for an example of a Cauchy sequences of rational numbers that does not have a rational limit.
	www.encyclopedian.com /ca/Cauchy-sequence.html (297 words)

	Real analysis/Sequences - Wikibooks, collection of open-content textbooks
		A further important property of sequences (arguably the most important property from the perspective of analysis) is the property of convergence.
		, a subsequence of this sequence is a sequence
		This is useful because, when the sequences converge, taking the limits commutes with the algebraic operation --- in other words, the limit of a sum of two sequences is equal to the sum of the individual limits, and similarly for the other operations.
	en.wikibooks.org /wiki/Real_analysis/Sequences (1462 words)

	Cauchy Sequences (Site not responding. Last check: 2007-10-07)
		A sequence s in a metric space is Cauchy (biography) if every ε implies an n such that all points beyond s
		Set ε to 1 to show that a Cauchy sequence, or the tail of a Cauchy function, is bounded.
		In other words, a sequence is Cauchy iff it is convergent.
	www.mathreference.com /top-ms,cauchy.html (241 words)

	Cauchy Sequences of Rationals
		This is concise notation for the Cauchy sequence 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, etc. Each term in the sequence brings in another decimal digit, an increase in precision, and the limit is sqrt(2).
		Let two sequences be equivalent if their difference is a sequence that approaches the rational number 0.
		The second sequence dominates the first, and is at least as large as the first.
	www.mathreference.com /top-ms,real.html (1518 words)

	Karl's Calculus Tutor: 1.0 Number Systems
		If the divisor sequence ends in all zeros, then the divisor is zero, and you can't divide by it (the same is true if the divisor converges to a limit of zero).
		Out of the sequence of sequences, take the first term of the first sequence, the second term of the second sequence, the third of the third, and so on.
		Because each sequence converges to something closer to the square root of two than the last, and because you can find one that converges to something as close to the square root of two as you'd like by going down the rows deeply enough.
	www.karlscalculus.org /calc1.html (3503 words)

	[No title]
		Theorem 10.5: A sequence that is bounded and eventually monotone converges.
		This says that the set of distinct points of the sequence is finite and thus no neighborhoods of any point can contain infinitely many distinct points of the sequence.
		This contradicts the notion that the sequence is Cauchy.
	www.puc.edu /faculty/George_Hilton/id94_m.htm (1056 words)

	Completeness Theorem in R
		The sequence is Cauchy if and only if it converges to some limit a.
		Second, assume that the sequence is Cauchy (this direction is much harder).
		Since the sequence is bounded (by part one of the theorem), say by a constant M, we know that every term in the sequence is bigger than -M.
	pirate.shu.edu /projects/reals/numseq/proofs/cauconv.html (217 words)

	Cauchy sequences (Site not responding. Last check: 2007-10-07)
		A sequence is called a Cauchy sequence if the terms of the sequence eventually all become arbitrarily close to one another.
		For example, the divergent sequence of partial sums of the harmonic series (see this earlier example) does satisfy this property, but not the condition for a Cauchy sequence.
		The fact that in R Cauchy sequences are the same as convergent sequences is sometimes called the Cauchy criterion for convergence.
	turnbull.mcs.st-and.ac.uk /~john/analysis/Lectures/L10.html (452 words)

	No Title (Site not responding. Last check: 2007-10-07)
		We may restate this theorem in the context of sequences.
		The point c is a cluster point of the set S if and only if there is a sequence of elements of S, all different from c, and converging to c.
		Our previous result may then be stated as follows: A convergent sequence is a Cauchy sequence.
	www.math.sunyit.edu /math/mat425/lectures/lecture16/lecture16.html (260 words)

	3.2. Cauchy Sequences (Site not responding. Last check: 2007-10-07)
		What is slightly annoying for the mathematician (in theory and in praxis) is that we refer to the limit of a sequence in the definition of a convergent sequence when that limit may not be known at all.
		Thus, by considering Cauchy sequences instead of convergent sequences we do not need to refer to the unknown limit of a sequence, and in effect both concepts are the same.
		Therefore, the rational numbers are not complete, in the sense that not every Cauchy sequence of rational numbers converges to a rational number.
	web01.shu.edu /projects/reals/numseq/causeq.html (288 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Uniform convergence implies properties for the limit of a sequence of functions.
		Cauchy's Residue Theorem is helpful to compute difficult definite integrals.
		In the realm of real numbers, proving that a sequence converges and proving it's a Cauchy sequence are just two aspects of the same thing.
	home.att.net /~numericana/answer/analysis.htm (3995 words)

	sequences 11.nb
		A sequence that is not finite is an infinite sequence.
		Any sequence whose higher order terms are smaller than previous terms is called a Cauchy sequence (after the famous mathematician Augustin Louis Cauchy, 1789-1857).
		It is easy to see that the above example of the sequence representing 1/3 is a Cauchy sequence, as each term of the sequence is progressively smaller than its predecessor.
	www.sas.org /E-Bulletin/2003-01-17/mathCorner/body.html (550 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		A real number is an equivalence class of Cauchy sequences of rationals, where a sequence x1,x2,...
		Every Cauchy sequence in the p-adic norm is equivalent to one where each term has a finite base p expansion.
		These two elements happen to be the limit of a sequence of (ordinary) integers, and we call elements of Q_p which are the limit of integers "p-adic integers", and denote the set of all such p-adics by Z_p.
	www.math.niu.edu /~rusin/papers/known-math/98/p-adics (1090 words)

	Augustin-Louis Cauchy (Site not responding. Last check: 2007-10-07)
		Even though this criterion for convergence bears Cauchy's name, he was not the first mathematician to derive this condition.
		It is only after these definitions for convergent and divergent series that Cauchy was able to state his criterion for convergence.
		What Cauchy did do was present examples where his criterion holds true.
	math.berkeley.edu /~robin/Cauchy/convergence.html (507 words)

	Biography of Cauchy (Site not responding. Last check: 2007-10-07)
		In 1812 Cauchy investigated symmetric functions and submitted a memoir later published in the Ecole Polytechnique in 1815.
		Although Cauchy was elected, because he did not take the oath he was not allowed to attend any meetings or receive a salary.
		Many terms in mathematics bare his name, the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations, and the Cauchy sequences.
	www.andrews.edu /~calkins/math/biograph/biocauch.htm (1433 words)

	the auroran sunset diary - the "constructive reals calculator"
		a cauchy sequence q is defined as a sequence such that "for all possible integers [e.g.
		so the definition is saying that with a cauchy sequence, no matter how small an error margin you decide is acceptable, you can find a member of the sequence so that the difference between all later pairs from the sequence is strictly less than that error.
		then the conditions for a cauchy sequence are met.
	www.livejournal.com /users/tithonus/319282.html (1485 words)

	Karl's Calculus Tutor - Proof of Real Limit of Cauchy Sequences of Real Numbers
		In the main text we discussed how a Cauchy sequence of real numbers is really a Cauchy sequence of Cauchy sequences.
		And each of those sequences is a sequence of rational numbers.
		sequence we have constructed is indeed a Cauchy sequence.
	www.karlscalculus.org /cauchyproof.html (696 words)

	Uniform continuity - RecipeFacts (Site not responding. Last check: 2007-10-07)
		) is a Cauchy sequence and f is a uniformly continuous function, then (f(x
		The most natural and general setting for the study of uniform continuity are the uniform spaces.
		In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences and that continuous maps on compact uniform spaces are automatically uniformly continuous.
	www.recipeland.com /encyclopaedia/index.php/Uniform_continuity (469 words)

	PlanetMath: Cauchy sequence (Site not responding. Last check: 2007-10-07)
		is a Cauchy sequence if, for every real number
		is a Cauchy sequence if and only if for every neighborhood
		This is version 5 of Cauchy sequence, born on 2001-10-27, modified 2005-11-30.
	planetmath.org /encyclopedia/CauchySequence.html (93 words)

	Cauchy Criterion for Series (Site not responding. Last check: 2007-10-07)
		} is a Cauchy sequence, and hence convergent.
		Then, by definition, the sequence of partial sums converges.
		But that, in turn, means that the Cauchy criterion for series holds.
	pirate.shu.edu /projects/reals/numser/proofs/caucrit.html (89 words)

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